Local Langlands in Families

朗兰兹程序是一组广泛的猜想，为理解L函数的神秘性质提供了一条途径，该函数编码了关于多项式方程的解的信息。这些猜想激发了代数论领域的广泛研究，并揭示了长期存在的悬而未决的问题。在某些情况下，朗兰兹猜想已被证明是正确的，但仍有一些微妙之处鲜为人知。在其他情况下，它们甚至还没有得到准确的表述。本项目建议在几何变形的背景下研究朗兰兹方案。除了数论研究目标外，该项目还包括一个针对教育工作者的培训部分：在该项目的每一年中，为缅因州各地的高中教师，特别是为农村地区服务的高中教师提供为期三周的夏季数论课程。作为该计划的一部分，教育工作者将为当地高中开发数论的课外活动或选修课。该项目的第一个研究目标是在家族中建立所谓的局部朗兰兹猜想方面取得进展，该猜想假设一个假定的局部朗兰兹对应升级到该对应两侧积分对象的模空间之间的映射，假设有一个简短的期望列表。第二个目标是研究族中的局部朗兰兹猜想与最近围绕局部朗兰兹猜想的p-进几何化的猜想之间的逻辑依赖关系。该方法是将逆定理和Plcherel测度推广到族中，以推广以前在一般线性群和经典群的特例中使用的一条论据。该项目由代数和数论计划和既定的激励竞争性研究计划(EPSCoR)共同资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The Langlands program is a broad set of conjectures that provide an avenue toward understanding the mysterious properties of L-functions, which encode information about solutions to polynomial equations. These conjectures have motivated broad swaths of research in the field of algebraic number theory and have shed light on long-standing open problems. In some contexts, the Langlands conjectures have been proven true, but there remain subtleties that are poorly understood. In other contexts, they are yet to even be precisely formulated. This project proposes to study the Langlands program within the context of geometric deformation. In addition to the number theory research goals, the project also includes a training component for educators: a three-week summer number theory program during each year of the project for high school teachers throughout Maine, especially those serving rural areas. As part of the program, educators will develop extracurricular activities or elective courses in number theory for their local high school. The first research goal of the project is to make progress toward establishing the so-called local Langlands conjecture in families, which upgrades a presumed local Langlands correspondence to a map between moduli spaces of integral objects on either side of the correspondence, assuming a short list of desiderata. The second goal is to study the logical dependence between the local Langlands conjecture in families and the conjectures surrounding recent p-adic geometrizations of the local Langlands conjectures. The method is to generalize converse theorems and the Plancherel measure in families in order to extend a line of argument previously used in the special cases of general linear groups and classical groups. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.